Can we partition all positive real numbers that are greater or equal to one, i.e. $[1, \infty)$ into sets of $[n, n+1)$ such that $n$ is a natural number?
Intuitively it is obviously correct but is there a way to prove it by using archimedean property?
Assume 1 <= r. Prove B = { n in N : r < n } is non empty.
Next show B has minimum element. Call that integer n.
As 1 <= r < n, conclude 1 <= n-1 <= r < n.